However, the utility function is not unique. Suppose a set of utilities accurately reflects a player’s preferences over outcomes. Let u be the individual’s utility for an outcome, a > 0, and b be any real number. Then converting all utilities by the transformation au + b maintains identical preferences.

Consequently, if we take a player’s utilities from a game and convert them all by au + b, the equilibria will be the same across the games. This remains true even if we change both players’ utilities, using a different a and b for each player.

Interestingly, the au + b transformation is the only transformation that preserves equilibria in this manner.